Quantum field theoretical methods in the theory of …utf.mff.cuni.cz/~scholtz/data/qft-turb.pdf ·...
Transcript of Quantum field theoretical methods in the theory of …utf.mff.cuni.cz/~scholtz/data/qft-turb.pdf ·...
Quantum field theoretical methods in the theoryof turbulence
Martin Scholtz
Institute of Theoretical PhysicsCharles University in Prague
May 31st, Ondrejov
Overview
1 Kolmogorov theoryFully developed turbulence
2 Quantum field theory
3 QFT and stochastic dynamics
4 Without conclusion
Transition from laminar to turbulent flow
Flow past the cylinderSmall Reynolds numbers (R = 0.16)
Up-down symmetryLeft-right symmetry (not exact)t−invariance (stationarity)z−invariance (2D character of the flow)
Transition from laminar to turbulent flow
Flow past the cylinderHigher Reynolds numbers (R ≈ 40)
Left-right symmetry completely brokenStationarity replaced by periodicity (Andronov-Hopfbifurcation)Continuous t−invariance replaced by discrete one
Transition from laminar to turbulent flow
Flow past the cylinderEven higher Reynolds numbers (R ≈ 140)
Breakdown of z−symmetry somewhere between R = 40 andR = 70Formation of Karman streets
Transition from laminar to turbulent flow
Transition to turbulent flow is connected with gradual loss ofsymmetries of the initial flow
However, as the flow becomes more and more chaotic, evenhigher symmetries are recovered in the limit R→∞
R = 240 R = 1800
Fully developed turbulence
Fully developed turbulence (FDT) for R→∞
Translational and rotational invariance
Stationarity, homogeneity and isotropy in statistical sense
Kolmogorov theory
Phenomenological picture of FDT
Three scales
Dissipation scale rdDissipation of energy by viscosityMolecular scale
External scale rlScale of energy injectionCreation of large eddies
Inertial range rd � r � rl
Transfer of energy from large scales to the smaller onesGoverned by non-linearity of the NS equation
Kolmogorov theory
Only statistical description is of interest
Corellation tensor
Sij = 〈vi(x1) vj(x2)〉
Structure functions
SN =⟨
[vr(x1)− vr(x2)]N⟩
In appropriate frame〈vi(x)〉 = 0
Homogeneity, isotropy and stationarity means
SN = SN (|x1 − x2|)
Kolmogorov theory
Characteristic parameters of fully developed turbulence
Dissipation rate or energy injection rate
ε =energy
time×mass, [ε] = L2 T−3
The “mass” (lower bound for wave number k)
m =1
rl, [m] = L−1
Inverse dissipation scale (upper bound for k, not acosmological constant!)
Λ =1
rd, [Λ] = L−1
Kolmogorov theory
Kolmogorov hypotheses
First hypothesis
H1. The properties of FDT in the inertial range do not depend onthe details of energy injection, only on the total energy injectionrate ε.
In other words, SN does not depend on rl,m
Second hypothesis
H2. The properties of FDT in the inertial range do not depend onthe details of the dissipation.
Hence, SN does not depend on rd, ν
Kolmogorov theory
Dimension of the structure function
[SN ] = LN T−N
Inside the inertial range rd � r � rl
Scale-invariant lawSN ∝ (ε r)N/3
Kinetic energy per unit mass
E =1
2〈vi vi〉 =
∞∫0
E(k) dk, [E(k)] = L3 T−2
Kolmogorov “5/3” law
E(k) = K0 ε2/3 k−5/3
Drawbacks of Kolmogorov theory
Experimental deviations from “5/3” lawadd experimental figure
Experiments show dependence on rl
SN = (ε r)N/3(r
rl
)qNAnomalous scaling
qN < 0 → limr/rl→0
SN =∞
Path integrals: a brief review
Quantum mechanics in 1D
Propagator is the amplitude of probablity of
|q0〉 at time 0 −→ |q1〉 at time t
Definition
K(q1, t; q0, 0) =⟨q1 | e−i t H | q0
⟩Path integral expression
K(q1, t; q0, 0) =
∫Dq(t) ei S[q]
where S[q] is classical action
S[q] =
∫ [1
2m q2 − V (q)
]dt
Quantum field theory
Action for free scalar field
S0[φ] =
∫ [1
2(∂µφ) (∂µφ)− 1
2m2 φ2
]d4x
Green’s correlation functions
G(n)(x1, . . . xn) =⟨
0 | Tφ(x1) · · · φ(xn) | 0⟩
Path integral expression
G(n)(x1, . . . xn) =1
Z
∫φ(x1) . . . φ(xn) eiS[φ]Dφ
Quantum field theory
Generating functional for Green functions
G0[J ] =
∫eiS0[φ]+iJφDφ∫eiS0[φ]Dφ , Jφ ≡
∫φ(x) J(x) d4x
Generating functional
G0[J ] = eiS0[φc]+iJφc = e−12J ∆F J
where φc is classical solution of (� +m2)φc = J :
φc(x) = i
∫∆F (x, x′) J(x′) d4x′
Quantum field theory
Green functions can be derived from G0
G(n)(x1, . . . xn) =1
inδnG0[J ]
δJ(x1) . . . δJ(xn)
∣∣∣∣J=0
Taylor expansion
G0[J ] = e−12J∆F J = 1− 1
2J ∆F J +
1
8(J ∆F J)2 + . . .
One-point Green function
G(1)(x) =⟨
0 | φ(x) | 0⟩
=δG0
δJ(x)
∣∣∣∣J=0
= 0
Quantum field theory
Two-point Green function
G(2)(x1, x2) =⟨
0 | φ(x1) φ(x2) | 0⟩
= ∆F (x1, x2)
or diagramatically
G(2)(x1, x2) = x1 x2
Quantum field theory
Four-point Green function (free propagator)
G(4)(x1, x2, x3, x4) = ∆F (x1, x2)∆F (x3, x4)+
∆F (x1, x3)∆F (x2, x4) + ∆F (x1, x4)∆F (x2, x3)
or diagramatically
G(4)(x1, x2, x3, x4) =
x1 x2
x3 x4
x1 x2
x3 x4
x1 x2
x3 x4
+ +
Quantum field theory
Self-interacting scalar field
S[φ] =
∫ [1
2(∂µφ)(∂µφ)− 1
2m2 φ2 − λ
4!φ4
]2-point Green’s function (to linear order in λ)
G(2) = ∆F (x1, x2)− i λ2
∫∆F (x1, x) ∆F (x, x) ∆F (x, x2) dx
Diagramatically
G(2)(x1, x2) =x1 x2 x1 x x2
+
Multicomponent fields
Example: charged scalar field
Two-component scalar field
Φ = (φ, φ∗)
Action of the free field
S0[Φ] = (∂µφ)(∂µφ∗)−m2 φφ∗
Alternative way of writing the action
S0[Φ] = −1
2Φ M Φ,
where
M =
(0 � +m2
� +m2 0
)
Multicomponent fields
Generating functional
G0[J ] =1
Z
∫DΦ e−
i2
ΦMΦ+iJΦ
Again we introduce classical field Φc and find
G0[J ] = e−12
ΦcMΦc+iJΦc
where classical equations of motion are
MαβΦβ = Jα
{(� +m2)φ = Jφ(� +m2)φ∗ = Jφ∗
Multicomponent fields
Solution of classical eqation
Green function – matrix ∆αβ
Mαβ∆βγ(x, x′) = −i δαγ δ(x− x′)
Solution
Φcα = i
∫∆αβ(x, x′) Jβ(x′) dx′
Generating functional
G0[J ] = e−12J∆J
where
−1
2J∆J = −1
2
∫d4x
∫d4x′ Jα(x)∆αβ(x, x′)Jβ(x′)
Multicomponent fields
Summary
Free action is written in the form
S0[Φ] = −1
2ΦMΦ
The matrix of free propagators is ∆ defined by
Mαβ∆βγ(x, x′) = −i δαγ δ(x− x′),
i.e. ∆ is the “inverse” of M
QFT formulation
Equivalent QFT problem
Stochastic problem given by differential equation
∂tφ = U(x, φ(x)) + η,
where η is stochastic noise with
〈η〉 = 0,
〈η(x) η(x′)〉 = D(x, x′),
is equivalent to quantum field theory problem with the action
S[φ, φ′] =1
2φ′Dφ′ + φ′ [−∂tφ+ U(x, φ(x))]
QFT formulation
Stochastic incompressible Navier-Stokes equation
∂tvi + vj∂jvi = −∂iP + ν∆vi + fi
f is stochastic stirring force
Incompressibility implies
P = −∂i∂j∆
(vivj)
so that the transversal part of Navier-Stokes equation is
∂tvi + vj∂jvi = ν∆vi + fi
QFT formulation
QFT formulation of “free” Navier-Stokes equation
S0[v′,v] =1
2v′Dv′ + v′ [−∂tv + ν∆v]
v′ is an auxiliary, unphysical field
D ≡ Dij(x, x′) is the corellator of stochastic force
Interaction term
SI [v′,v] = −v′ (v · ∇v)
QFT formulation
Free action is written in the form
S0[Φ] = −1
2ΦM Φ,
where Φ = (v,v′),
M = P(
0 −∂t − ν∆∂t − ν∆ −D
)P
and P is transversal projector
QFT formulation
In the Fourier space
M =
(0 (−i ω + ν k2)Pαβ
(i ω + ν k2)Pαβ −PαµDµνPνβ
)where Pαβ = δαβ − kα kβ
k2
Matrix of free propagators is defined by
Mαβ ∆βγ = Pαγ
QFT formulation
Free propagators for Navier-Stokes equation
=PαµDµνPνβω2+ν k4〈vα vβ〉0 = ∆vv
αβ =vα vβ
=Pαβ
−i ω+ν k2〈vα v′β〉0 = ∆vv′αβ =
vα v′β
=Pαβ
i ω+ν k2〈v′α vβ〉0 = ∆v′vαβ =
v′α vβ
= 0〈v′α v′β〉0 = ∆v′v′αβ =
v′α v′β
QFT formulation
Interaction termSI = −v′ (v · ∇v)
In the Fourier space
Vαβµ = i kβ δαµ
Diagramatically
v′α
vβ
vµ
Vαβµ = = i kβ δαµ
QFT formulation
Generating functional
G[J ] =1
Z
∫dΦ ei S0[Φ]+i SI [Φ]+i JΦ
1-loop approximation
〈vv〉 = +
+ 12
QFT formulation
Explicit form of the Feynman diagram
vα v′γp
k
p− kp
vδ
vρ
v′λ
vσvβvµ
Σvvαβ =
Σvvαβ(p) =
1
(2π)4
∫d4k ∆vv′
αγ (p)Vγδρ(p) ∆vv′δλ (k)×
×∆vvρσ(p− k)Vλσµ(k) ∆vv
µβ(p)
Exercise: compute the integral!
Correlator of stochastic force
Prescribed statistics for Fi⟨Fα(x)Fβ(x′)
⟩= Dαβ(x, x′)
Isotropy + transversality of velocity field + white noise
Dαβ(x, x′) =δ(t− t′)
(2π)2
∫dk Pαβ(k) dF (|k|) eik·(x−x′)
Mean energy injection
W ∝∫dF (k) dk
Correlator of stochastic force
Recall: inertial range is m� k � Λ
Deviation from Kolmogorov – structure functions depend onm
Energy injection should be infra-red, i.e. the maincontribution to W is from scales
k ∼ m
Physically realistic model
dF (k) = D0 k4−d(k2 +m2)−ε
ε is the small parameter of perturbative expansion
Without conclusion
What can be investigated?
Structure functions and scaling laws
Renormalised quantities
DiffusivityKolmogorov constant (related to energy spectrum)
Corrections to Kolmogorov laws
Explanation of anomalous scaling
Influence of anisotropy, compressibility and helicity
Simpler models
Analysis of Navier-Stokes equation is very difficultAdvection of passice admixturesScalar, vector admixture
. . .